Interactive Generative Manifold Learning

Nonlinear manifold learning is an active area of mathematical research. Unfortunately, the extant literature has far less to offer on the problem of interactive nonlinear manifold learning. In other words, satisfactory nonlinear manifold learning approaches that put the''human in the loop'are yet to be fully developed. Human steering of such calculations promises several advantages including, leveraging human expertise in sparse data environments, maximizing efficiency by allowing computational resources to be focused on areas of interest to the user, and augmenting the amount of useful information the user can glean from large and complicated data sets. Several questions immediately present themselves. What if the data is too voluminous to be processed all at once? What if one does not have all possible data at hand and must decide what additional data would be the most informative to synthesize, or to acquire? How does one best take advantage of the user's expertise and inject it into the problem? These considerations lead one inexorably to five core interactions between the user and the manifold learning algorithm that are not fully addressed in current manifold learning algorithms, namely: interpolation, extension, resampling, extrapolation, and visualization of the data by the user.